A Friendly Approach To Functional Analysis

by Amol Sasane

Overview

'The book is unusual among functional analysis books in devoting a lot of space to the derivative. The "˜friendly' aspect promised in the title is not explained, but there are three things I think would strike most students as friendly: the slow pace, the enormous number of examples, and complete solutions to all the exercises.'
MAA ReviewsThis book constitutes a concise introductory course on Functional Analysis for students who have studied calculus and linear algebra. The topics covered are Banach spaces, continuous linear transformations, Frechet derivative, geometry of Hilbert spaces, compact operators, and distributions. In addition, the book includes selected applications of functional analysis to differential equations, optimization, physics (classical and quantum mechanics), and numerical analysis. The book contains 197 problems, meant to reinforce the fundamental concepts. The inclusion of detailed solutions to all the exercises makes the book ideal also for self-study.A Friendly Approach to Functional Analysis is written specifically for undergraduate students of pure mathematics and engineering, and those studying joint programmes with mathematics.

Product Details

ISBN-13:	9781786343338
Publisher:	World Scientific Publishing Europe Ltd
Publication date:	04/19/2017
Series:	Essential Textbooks In Mathematics
Pages:	396
Product dimensions:	6.00(w) x 9.10(h) x 1.00(d)

Table of Contents

Preface vii

1 Normed and Banach spaces 1

1.1 Vector spaces 3

1.2 Normed spaces 7

1.3 Topology of normed spaces 17

1.4 Sequences in a normed space; Banach spaces 24

1.5 Compact sets 44

2 Continuous and linear maps 53

2.1 Linear transformations 54

2.2 Continuous maps 58

2.3 The normed space CL(X,Y) 67

2.4 Composition of continuous linear transformations 82

2.5 (*) Open Mapping Theorem 92

2.6 Spectral Theory 97

2.7 (*) Dual space and the Hahn-Banach Theorem 104

3 Differentiation 117

3.1 Definition of the derivative 118

3.2 Fundamental theorems of optimisation 125

3.3 Euler-Lagrange equation 134

3.4 An excursion in Classical Mechanics 145

4 Geometry of inner product spaces 155

4.1 Inner product spaces 156

4.2 Orthogonality 165

4.3 Best approximation 174

4.4 Generalised Fourier series 183

4.5 Riesz Representation Theorem 189

4.6 Adjoints of bounded operators 190

4.7 An excursion in Quantum Mechanics 200

5 Compact operators 209

5.1 Compact, operators 210

5.2 The set K(X,Y) of all compact operators 211

5.3 Approximation of compact operators 217

5.4 (*) Spectral Theorem for Compact Operators 222

6 A glimpse of distribution theory 227

6.1 Test functions, distributions, and examples 230

6.2 Derivatives in the distributional sense 237

6.3 Weak solutions 243

6.4 Multiplication by C^∞ functions 248

6.5 Fourier transform of (tempered) distributions 253

Solutions 259

The Lebesgue integral 359

Bibliography 373

Index 375